Tractable Approximate Robust Geometric Programming

The optimal solution of a geometric program (GP) can be sensitive to variations
in the problem data. Robust geometric programming can systematically alleviate
the sensitivity problem by explicitly incorporating a model of data uncertainty
in a GP and optimizing for the worst-case scenario under this model. However,
it is not known whether a general robust GP can be reformulated as a tractable
optimization problem that interior-point or other algorithms can efficiently
solve. In this paper we propose an approximation method that seeks a compromise
between solution accuracy and computational efficiency. The method is based on
approximating the robust GP as a robust linear program (LP), by replacing each
nonlinear constraint function with a piecewise linear (PWL) convex
approximation. With a polyhedral or ellipsoidal description of the uncertain
data, the resulting robust LP can be formulated as a standard convex
optimization problem that interior-point methods can solve. The drawback of
this basic method is that the number of terms in the PWL approximations
required to obtain an acceptable approximation error can be very large. To
overcome the curse of dimensionality that arises in directly approximating
the nonlinear constraint functions in the original robust GP, we form a
conservative approximation of the original robust GP, which contains only
bivariate constraint functions. We show how to find globally optimal PWL
approximations of these bivariate constraint functions.